Basis for the tensor product/Statement

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Let [ilmath]\mathbb{F} [/ilmath] be a field and let [ilmath]\big((V_i,\mathbb{F})\big)_{i\eq 1}^k[/ilmath] be a family of finite dimensional vector spaces. Let [ilmath]n_i:\eq\text{Dim}(V_i)[/ilmath] and [ilmath]e^{(i)}_1,\ldots,e^{(i)}_{n_i} [/ilmath] denote a basis for [ilmath]V_i[/ilmath], then we claim[1]:

  • [math]\mathcal{B}:\eq\left\{e^{(1)}_{i_1}\otimes\cdots\otimes e^{(k)}_{i_k}\ \big\vert\ \forall j\in\{1,\ldots,k\}\subset\mathbb{N}[1\le i_j\le n_j]\right\} [/math]

Is a basis for the tensor product of the family of vector spaces, [ilmath]V_1\otimes\cdots\otimes V_k[/ilmath]

Note that the number of elements of [ilmath]\mathcal{B} [/ilmath], denoted [ilmath]\vert\mathcal{B}\vert[/ilmath], is [ilmath]\prod_{i\eq 1}^kn_i[/ilmath] or [ilmath]\prod_{i\eq 1}^k\text{Dim}(V_i)[/ilmath], thus:

  • [ilmath]\text{Dim}(V_1\otimes\cdots\otimes V_k)\eq\prod_{i\eq 1}^k n_i[/ilmath][1]


  1. 1.0 1.1 Introduction to Smooth Manifolds - John M. Lee